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\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra pdecomp.spad}
\author{The Axiom Team}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{package PCOMP PolynomialComposition}
<<package PCOMP PolynomialComposition>>=
)abbrev package PCOMP PolynomialComposition
++ Description:
++ This package \undocumented
PolynomialComposition(UP: UnivariatePolynomialCategory(R), R: Ring): with
compose: (UP, UP) -> UP
++ compose(p,q) \undocumented
== add
compose(g, h) ==
r: UP := 0
while g ~= 0 repeat
r := leadingCoefficient(g)*h**degree(g) + r
g := reductum g
r
@
\section{package PDECOMP PolynomialDecomposition}
<<package PDECOMP PolynomialDecomposition>>=
)abbrev package PDECOMP PolynomialDecomposition
++ Description:
++ This package \undocumented
-- Ref: Kozen and Landau, Cornell University TR 86-773
PolynomialDecomposition(UP, F): PDcat == PDdef where
F:Field
UP:UnivariatePolynomialCategory F
NNI ==> NonNegativeInteger
LR ==> Record(left: UP, right: UP)
PDcat == with
decompose: UP -> List UP
++ decompose(up) \undocumented
decompose: (UP, NNI, NNI) -> Union(LR, "failed")
++ decompose(up,m,n) \undocumented
leftFactor: (UP, UP) -> Union(UP, "failed")
++ leftFactor(p,q) \undocumented
rightFactorCandidate: (UP, NNI) -> UP
++ rightFactorCandidate(p,n) \undocumented
PDdef == add
leftFactor(f, h) ==
g: UP := 0
for i in 0.. while f ~= 0 repeat
fr := divide(f, h)
f := fr.quotient; r := fr.remainder
positive? degree r => return "failed"
g := g + r * monomial(1, i)
g
decompose(f, dg, dh) ==
df := degree f
dg*dh ~= df => "failed"
h := rightFactorCandidate(f, dh)
g := leftFactor(f, h)
g case "failed" => "failed"
[g::UP, h]
decompose f ==
df := degree f
for dh in 2..df-1 | df rem dh = 0 repeat
h := rightFactorCandidate(f, dh)
g := leftFactor(f, h)
g case UP => return
append(decompose(g::UP), decompose h)
[f]
rightFactorCandidate(f, dh) ==
f := f/leadingCoefficient f
df := degree f
dg := df quo dh
h := monomial(1, dh)
for k in 1..dh repeat
hdg:= h**dg
c := (coefficient(f,(df-k)::NNI)-coefficient(hdg,(df-k)::NNI))/(dg::F)
h := h + monomial(c, (dh-k)::NNI)
h - monomial(coefficient(h, 0), 0) -- drop constant term
@
\section{License}
<<license>>=
--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
--All rights reserved.
--
--Redistribution and use in source and binary forms, with or without
--modification, are permitted provided that the following conditions are
--met:
--
-- - Redistributions of source code must retain the above copyright
-- notice, this list of conditions and the following disclaimer.
--
-- - Redistributions in binary form must reproduce the above copyright
-- notice, this list of conditions and the following disclaimer in
-- the documentation and/or other materials provided with the
-- distribution.
--
-- - Neither the name of The Numerical ALgorithms Group Ltd. nor the
-- names of its contributors may be used to endorse or promote products
-- derived from this software without specific prior written permission.
--
--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
@
<<*>>=
<<license>>
--% Polynomial composition and decomposition functions
-- If f = g o h then g = leftFactor(f, h) & h = rightFactor(f, g)
-- SMW Dec 86
<<package PCOMP PolynomialComposition>>
<<package PDECOMP PolynomialDecomposition>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
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